This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A349841 #9 Jul 31 2022 15:56:56
%S A349841 1,1,0,1,1,0,1,2,1,0,1,3,3,1,0,1,4,6,4,1,1,1,5,10,10,5,2,0,1,6,15,20,
%T A349841 15,7,2,0,1,7,21,35,35,22,9,2,0,1,8,28,56,70,57,31,11,2,0,1,9,36,84,
%U A349841 126,127,88,42,13,2,1
%N A349841 Triangle T(n,k) built by placing all ones on the left edge, [1,0,0,0,0] repeated on the right edge, and filling the body using the Pascal recurrence T(n,k) = T(n-1,k) + T(n-1,k-1).
%C A349841 This is the m=5 member in the sequence of triangles A007318, A059259, A118923, A349839, A349841 which have all ones on the left side, ones separated by m-1 zeros on the other side, and whose interiors obey Pascal's recurrence.
%C A349841 T(n,k) is the (n,n-k)-th entry of the (1/(1-x^5),x/(1-x)) Riordan array.
%C A349841 For n>0, T(n,n-1) = A002266(n+4).
%C A349841 For n>1, T(n,n-2) = A008732(n-2).
%C A349841 For n>2, T(n,n-3) = A122047(n-1).
%C A349841 Sums of rows give A349842.
%C A349841 Sums of antidiagonals give A349843.
%H A349841 Michael A. Allen and Kenneth Edwards, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL25/Allen/allen3.html">On Two Families of Generalizations of Pascal's Triangle</a>, J. Int. Seq. 25 (2022) Article 22.7.1.
%F A349841 G.f.: (1-x*y)/((1-(x*y)^5)(1-x-x*y)) in the sense that T(n,k) is the coefficient of x^n*y^k in the series expansion of the g.f.
%F A349841 T(n,0) = 1.
%F A349841 T(n,n) = delta(n mod 5,0).
%F A349841 T(n,1) = n-1 for n>0.
%F A349841 T(n,2) = (n-1)*(n-2)/2 for n>1.
%F A349841 T(n,3) = (n-1)*(n-2)*(n-3)/6 for n>2.
%F A349841 T(n,4) = (n-1)*(n-2)*(n-3)*(n-4)/24 for n>3.
%F A349841 T(n,5) = C(n-1,5) + 1 for n>4.
%F A349841 T(n,6) = C(n-1,6) + n - 6 for n>5.
%F A349841 For 0 <= k < n, T(n,k) = (n-k)*Sum_{j=0..floor(k/5)} binomial(n-5*j,n-k)/(n-5*j).
%F A349841 The g.f. of the n-th subdiagonal is 1/((1-x^5)(1-x)^n).
%e A349841 Triangle begins:
%e A349841 1;
%e A349841 1, 0;
%e A349841 1, 1, 0;
%e A349841 1, 2, 1, 0;
%e A349841 1, 3, 3, 1, 0;
%e A349841 1, 4, 6, 4, 1, 1;
%e A349841 1, 5, 10, 10, 5, 2, 0;
%e A349841 1, 6, 15, 20, 15, 7, 2, 0;
%e A349841 1, 7, 21, 35, 35, 22, 9, 2, 0;
%e A349841 1, 8, 28, 56, 70, 57, 31, 11, 2, 0;
%e A349841 1, 9, 36, 84, 126, 127, 88, 42, 13, 2, 1;
%t A349841 Flatten[Table[CoefficientList[Series[(1 - x*y)/((1 - (x*y)^5)(1 - x - x*y)), {x, 0, 20}, {y, 0, 10}], {x, y}][[n+1,k+1]],{n,0,10},{k,0,n}]]
%Y A349841 Other members of sequence of triangles: A007318, A059259, A118923, A349839.
%Y A349841 Columns: A000012, A001477, A000217, A000292, A000332, A323228.
%Y A349841 Diagonals: A079998, A002266, A008732, A122047.
%K A349841 easy,nonn,tabl
%O A349841 0,8
%A A349841 _Michael A. Allen_, Dec 13 2021